# Chapter 8 – Methods of Analysis Lecture 11 by Moeen Ghiyas 14/04/2015 1.

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Chapter 8 – Methods of Analysis Lecture 11 by Moeen Ghiyas 14/04/2015 1

CHAPTER 8

Bridge Networks Y – Δ (T – π) and Δ to Y (π – T) Conversions

A configuration that has a multitude of applications DC meters & AC meters Rectifying circuits (for converting a varying signal to one of a steady nature such as dc) Wheatstone bridge (smoke detector ) and other applications A bridge network may appear in one of the three forms

The network of Fig (c) is also called a symmetrical lattice network if R 2 = R 3 and R 1 = R 4. Figure (c) is an excellent example of how a planar network can be made to appear non-planar

Solution by Mesh Analysis (Format Approach)

Solution by Nodal Analysis (Format Approach) Can we replace R 5 with a short circuit here?

Can we replace R 5 with a short circuit? Since V 5 = 0V, yes! From nodal analysis we can insert a short in place of the bridge arm without affecting the network behaviour Lets determine V R4 and V R3 to confirm validity of short ie V R4 must equal V R3 As before V R4 and V R3 = 2.667 V

Can we replace same R 5 with a open circuit? From mesh analysis we know I 5 = 0A, therefore yes! we can have an open circuit in place of the bridge arm without affecting the network behaviour (Certainly I = V/R = 0/(∞ ) = 0 A) Lets determine V R4 and V R3 to confirm validity of open circuit ie V R4 must equal V R3

The bridge network is said to be balanced when the condition of I = 0 A or V = 0 V exists Lets derive relationship for bridge network meeting condition I = 0 and V = 0

If V = 0 (short cct b/w node a and b), then V 1 = V 2 or I 1 R 1 = I 2 R 2

Two circuit configurations not falling into series or parallel configuration and making it difficult to solve without the mesh or nodal analysis are Y and Δ or (T and π). Under these conditions, it may be necessary to convert the circuit from one form to another to solve for any unknown qtys Note that the pi (π) is actually an inverted delta (Δ)

Conversion will normally help to solve a network by using simple techniques With terminals a, b, and c held fast, if the wye (Y) configuration were desired instead of the inverted delta (Δ) configuration, all that would be necessary is a direct application of the equations, which we will derive now If the two circuits are to be equivalent, the total resistance between any two terminals must be the same

Consider terminals a-c in the Δ -Y configurations of Fig

If the resistance is to be the same between terminals a-c, then To convert the Δ (RA, RB, RC) to Y (R1, R2, R3)

Note that each resistor of the Y is equal to the product of the resistors in the two closest branches of the Δ divided by the sum of the resistors in the Δ.

Note that the value of each resistor of the Δ is equal to the sum of the possible product combinations of the resistances of the Y divided by the resistance of the Y farthest from the resistor to be determined

what would occur if all the values of a Δ or Y were the same. If R A = R B = R C

The Y and the Δ will often appear as shown in Fig. They are then referred to as a tee (T) and a pi (π) network, respectively

Example – Find the total resistance of the network

Bridge Networks Y – Δ (T – π) and Δ to Y (π – T) Conversions

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